
// Make newform 8034.2.a.s in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8034_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-2, -15, -15, 68, 54, -115, -27, 72, -15, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-1728, -3054, 7872, 7648, -8561, -3919, 3250, -115, -168, 20], [-94, -5055, -1890, 18177, 2593, -14211, 1005, 1696, -68, -54], [3027, 10486, -9932, -27696, 11815, 17110, -6760, -804, 369, -9], [-1830, -4922, 6598, 16340, -12628, -11564, 10538, -1226, -599, 114], [-665, -1280, 5402, 7957, -13349, -6323, 11412, -2259, -651, 159], [-2322, -9167, 9111, 26903, -22542, -18189, 20331, -3114, -1163, 251], [1591, 6103, -14601, -17531, 33629, 8252, -23303, 5122, 1307, -338], [-2145, -5657, 17813, 18685, -43813, -8734, 31789, -7452, -1791, 481]];
    Rf_basisdens := [1, 1, 163, 163, 163, 163, 163, 163, 163, 163];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_8034_a_Hecke(Kf)
    return MakeCharacter_8034_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 1, 1, 1, 1, -1, 0, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 1], [0, -1, 0, 0, 1, 0, 0, -1, 0, 0], [1, 1, 0, -1, 1, 0, -1, -1, 0, 0], [-3, 0, -1, -1, -2, -2, 2, 0, 1, -1], [-2, 1, 0, 0, 1, 0, 0, 1, 0, 0], [1, 1, 1, -1, 0, -2, 2, 0, 1, 1], [-2, 0, 0, -1, -2, -1, 2, 0, 1, -1], [0, 1, 0, 1, -2, 1, -1, 0, -1, 0], [2, -3, -2, 0, -1, 1, 1, 0, -2, -2], [-2, 1, -1, 1, -1, -3, 1, 2, 1, 1], [1, -1, -1, -2, 0, 1, 0, 0, -2, -1], [-2, 1, 1, 1, -1, -2, -1, 2, 3, 3], [-1, 0, 1, 0, 2, 2, 1, -1, 0, 0], [-3, 0, -1, -1, -1, -4, 1, 1, 1, 0], [3, -1, 4, 1, 3, 2, -2, 0, -1, 3], [0, -3, -1, 0, -2, -2, 0, -1, 1, 1], [1, -1, 2, 2, 1, -1, 1, 1, -1, 0], [3, 2, 2, 0, 2, 3, -2, 0, -2, 2], [-2, 0, 1, 0, 1, -2, 0, 2, -1, 1], [1, 1, 2, 1, 0, -4, 0, 0, 0, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -3, 0, -1, 0, -1, -2, 1, -2, 2], [-5, 0, -2, -2, -2, -2, 1, 0, 1, -1], [0, 3, 4, 3, 2, -2, 0, 3, 2, 3], [-4, -1, -2, 0, -2, 1, 2, -2, 1, -1], [-2, 1, 0, 1, 3, 8, -2, -2, -6, -1], [1, 0, 0, 2, 0, 3, -1, -2, -2, 0], [-4, 0, -2, 0, -2, 0, 2, -1, 1, 0], [-3, -3, -4, -2, 1, 4, 3, -2, -2, -4], [-4, 0, 3, 3, 0, -5, 0, 1, 4, 3], [5, -1, -1, -3, 1, 0, -3, -1, -2, 1], [-4, 4, 4, 2, 4, 1, -4, 4, -2, 3], [0, 1, -1, 0, -3, -4, 0, 0, 0, 2], [4, 0, 4, 0, -2, -5, 1, 3, 0, 3], [6, -1, 5, 1, 4, 5, -5, 0, -4, 1], [-2, -3, -1, 1, -2, -2, -1, 2, 0, -2], [2, 4, 4, -1, 1, -1, -2, -1, 3, 2], [2, -3, -3, -2, 2, 7, 0, -4, -4, -6], [-5, -1, -4, -2, -1, 1, 0, -1, -2, -4], [0, -1, -1, -2, -1, 2, 0, 0, -3, 0], [-8, 1, 3, -3, 0, -4, 0, 0, 0, 0], [-11, 2, -4, 0, -3, -4, 4, 0, 4, -2], [5, 5, 1, -1, 3, 2, -1, -4, 3, 3], [4, -1, -5, -1, 2, 7, -3, -2, -3, -2], [0, 3, 1, 1, 2, 0, -2, 3, 3, 1], [-5, 3, -1, 2, 0, -4, 3, 0, 4, -1], [-9, 1, 2, 1, 1, -1, 0, 1, -2, 2], [1, 1, -4, 0, -3, 8, -4, -1, -3, 0], [3, -4, -2, 0, 2, 4, 1, -5, -1, -2], [-8, 1, 0, 2, 4, 5, 1, -1, -4, -3], [8, 0, 1, -1, -2, -3, 2, 1, 4, -1], [-14, -2, -6, 0, -2, -2, 0, -1, -1, -4], [-1, 1, 0, 0, -2, -6, 5, 1, 4, -1], [1, 4, -4, -5, -3, 1, 0, -3, -2, -4], [0, -2, 0, -1, -4, 0, -3, -1, 2, -1], [-3, 3, -1, 2, 1, 4, -5, 1, -1, 0], [-7, -1, -2, 0, 4, 8, -4, 0, -8, -7], [4, 1, -3, -1, 2, 0, 0, 1, -1, 1], [2, 5, 0, 3, -2, 3, -2, -1, 1, 0], [9, 0, 1, 2, 2, -1, -3, 4, 0, 0], [0, 1, -3, -2, -1, 0, -1, 1, 2, 0], [3, 5, 5, -1, -2, -3, 1, 0, 3, 2], [10, -2, -4, -3, -3, -3, 0, -1, 0, 3], [-4, -1, 1, 2, -3, 1, -1, 1, -3, 0], [1, 1, -7, 1, -7, 0, 1, -3, 2, -3], [1, 3, 3, 4, 5, 11, -2, 2, -8, -4], [-3, -4, -7, -4, -4, -7, 5, -4, 4, -4], [11, -4, -2, -3, 2, 0, 2, -1, -4, 1], [-5, -3, -1, -2, -5, -3, 5, 0, 1, 0], [2, 3, -2, 0, -2, 4, 1, -1, 0, -3], [-4, -1, -1, 0, 6, 0, 2, -3, 2, -4], [0, 2, -2, -2, 2, 2, 3, -2, 1, 2], [-6, -3, 1, 0, 4, 8, -1, 0, 0, 1], [-7, 7, -1, 1, 0, 3, -1, -3, 3, -4], [-3, 4, 3, 1, -7, -12, 3, 1, 6, 5], [-8, 7, -2, 3, 2, 9, 0, -1, -4, -6], [1, 1, -1, 1, 2, 8, -4, 0, -4, -2], [1, 3, 2, -1, 2, 7, -4, -1, -5, 4], [-21, 3, -2, 0, -3, 0, 1, 2, -2, 0], [5, -4, -5, -1, -3, -5, 3, 3, 2, 4], [-8, 0, 7, 4, 0, -6, 1, 7, 1, 2], [-10, -2, -2, 0, 0, 9, -3, -5, 0, -1], [-2, 0, 8, 4, 5, 8, -2, 2, -7, -1], [-11, 0, 3, 2, -3, -4, 0, 0, -2, -3], [2, -1, -6, -2, -4, -2, 9, -4, 3, -8], [-6, 0, 6, -1, -6, -17, 3, 6, 6, 5], [-3, -3, 3, -2, 2, -2, 1, 3, 3, -1], [-13, 2, 1, 0, 1, 1, -2, -2, -1, -1], [-7, -2, 2, 2, 0, 1, 0, 3, -3, 0], [10, 0, 1, 2, 1, 4, 3, 0, 3, 0], [-8, 3, 6, 3, 4, 4, -5, 5, 1, 6], [-2, 1, -1, 1, -1, -1, 0, 3, 0, 4], [-2, 1, 9, 3, 2, -3, -1, 3, -4, 2], [-5, 3, -9, -5, -6, -5, 1, -4, 4, -2], [-2, 8, -1, -6, 0, 2, -1, 0, 2, 2], [5, 6, 2, 5, 0, 7, -7, 1, 3, 7], [5, -1, -7, -3, 3, 9, 0, -3, -8, -3], [-10, -5, -10, -1, 0, 6, 2, -4, -3, -5], [-4, -2, 0, 1, 4, 5, 1, 0, -3, -5], [-12, 3, -4, -6, -4, -1, 2, -1, -4, -4], [0, 6, 3, 1, 4, 8, -6, -1, -9, -1], [13, 5, 3, 0, -2, -5, -3, 3, 8, 3], [-6, 2, 7, 6, 5, 4, 0, 3, 0, 3], [-19, -2, 0, -3, 2, -7, 1, -1, 1, 1], [2, -1, 6, 8, 3, -2, -7, 6, 4, 10], [1, 3, 8, 3, 12, 10, -10, 2, -3, 8], [3, -6, -11, -2, -6, 2, 4, -4, 2, -8], [-4, -4, -7, 2, -1, 6, 0, -1, -4, -1], [-7, 1, 7, -1, 6, -6, 1, 7, -3, 7], [-6, 3, 5, 5, 0, -15, 4, 6, 12, 6], [3, -5, -8, -3, -6, -1, 2, 1, 2, -7], [-2, 5, -3, 4, 0, 6, -4, 3, -2, -2], [-11, 8, 0, 4, -2, 3, -1, 1, 3, 1], [-1, -1, -14, -4, -9, -1, 2, -5, 8, -4], [5, 0, 4, -1, 7, -2, 1, -3, 0, 2], [-4, -8, 11, 3, 5, 3, -2, 2, -12, 2], [0, -2, 11, 5, 2, -9, 1, 9, 3, 9], [-1, 0, -12, -3, 2, 18, -4, -7, -5, -7], [-17, -5, -11, -1, -7, -2, 6, -3, 5, -5], [0, -7, 1, -3, -6, -4, 0, -3, 4, 2], [-8, 2, 2, 3, -1, 3, 2, -1, -3, 0], [3, -7, 1, 1, 0, 2, -5, -3, -4, 2], [-4, 1, -3, 1, -9, 5, -1, -1, 2, -1], [-1, 7, -1, -1, 3, 7, -1, -3, -8, -6], [-12, 0, -2, 5, -4, 1, -2, 1, 2, -6], [-10, 0, 1, 1, -3, -2, 1, 3, -6, 4], [-8, 3, 6, -1, 7, 4, -4, 4, -6, 5], [2, 6, -1, -1, 1, -2, 5, 0, 2, 0], [-1, 9, 10, 3, 2, -4, -9, 5, 1, 10], [7, -6, 6, 5, 4, -1, 1, 1, 2, 6], [6, 4, -1, -3, -3, -8, 4, 3, 2, -6], [-9, 3, -8, -3, 2, 2, 1, -6, 3, 1], [7, -9, -7, 2, -5, 4, 3, -2, -2, -8], [7, -6, 1, -5, 4, 4, -7, 2, -6, 3], [-13, 2, 1, -1, 0, -2, -3, -1, 5, 1], [0, -6, 5, -4, 6, -5, 3, -5, 3, 6], [1, 9, 9, 8, 6, -2, -3, 5, 8, 5], [12, 2, 8, -1, 8, 4, -1, 1, -2, 5], [-13, -1, -5, 2, 0, 5, -5, 0, -9, -7], [-8, -1, -1, -3, -4, -3, 0, 3, -3, -6], [0, 4, 2, -1, -9, -19, 11, 3, 10, 2], [-4, -1, 6, 0, 8, -1, -6, 3, -5, 5], [6, 0, -4, -2, 1, 6, -2, 3, -4, 1], [-6, -8, -12, -7, -3, -3, 2, -6, 3, -4], [-5, -1, -2, -4, -12, -9, 8, 6, -2, -4], [-4, -3, -2, 1, 0, -9, 6, 5, -2, -6], [-4, -1, -5, 1, 5, 3, 2, 1, 3, -5], [-7, -2, -5, -1, 4, 12, -3, -3, -4, -2], [-2, 2, -8, -3, -9, -2, 1, 0, 0, -6], [-23, 5, -12, -6, -12, -6, 9, -2, 5, -9], [1, -5, -7, 0, -5, -7, 2, -6, 5, -2], [-2, 7, -3, 4, -4, 6, -2, -4, 1, -5], [-11, 5, 6, -1, 6, -6, 2, 2, 5, 0], [-11, 8, 6, 2, -6, -9, 8, -2, 10, -2], [0, -6, -3, -5, -3, 0, 2, -5, 4, 4], [-7, 0, -14, -10, -4, -2, 3, -8, -8, -10], [-2, -2, -6, -11, -5, -14, 12, 2, 1, 0], [0, 10, 5, 0, 8, 3, -2, 2, 0, 0], [-10, -1, -7, 4, -7, -8, 8, 2, 9, 0], [-7, 3, -1, -8, -3, -16, 6, 2, 7, 0], [-4, -6, -12, -3, -7, 1, 2, -3, 1, 4], [-2, 0, 9, 1, -2, -3, -1, 9, -3, 1], [-18, -4, -6, 0, 0, 4, 3, 5, -10, -2], [12, -1, 7, -3, -1, -3, 3, 0, -2, 4], [-3, 4, 2, -3, 0, -10, 5, 2, 13, 7], [0, -13, 4, 5, 9, 15, -4, -4, -10, 2], [-16, 5, -3, -1, 3, 4, 6, -1, -6, -3], [10, -12, 1, -1, -9, -10, 2, -1, 8, -1], [-1, -8, -4, -2, -4, -3, 4, 1, 3, -6], [27, 6, 2, 0, 3, -5, 2, -1, 7, -5], [-18, -8, 0, -1, 1, -1, 6, -1, 3, 0], [12, -4, 12, 7, 7, 7, -4, 7, -1, 6], [-8, 2, -6, -1, -5, 6, 2, -6, -1, -8], [-17, 5, 13, 8, 7, -4, 2, 6, 3, 8], [5, 6, 6, 3, 7, 8, -8, -5, 3, 6], [-10, 10, -9, -1, -2, 8, 0, -7, 4, -8], [11, 3, -2, 7, -1, 10, 4, -1, -3, -3], [-5, 1, -3, 2, 4, 17, -1, -1, -4, -5], [2, -5, 2, 4, -4, -1, 3, -3, 12, -3], [-1, -1, 5, -2, -5, -10, 3, 1, 3, -2], [-2, -9, -3, 4, -6, 3, 0, 0, -3, 1], [-27, 6, -2, -1, 2, 1, 3, 4, -5, 0], [-15, 4, 1, -2, -6, -13, 2, 6, 9, 4], [-6, 6, 11, 4, 1, 4, 1, 4, 0, -1], [22, -7, -3, -4, -6, -2, 2, -11, 7, -1], [-3, 5, -11, -1, -7, -12, 8, -5, 11, -1], [-1, 11, 5, 10, 1, 17, -9, -1, -10, 4], [19, -14, 2, -3, 2, -6, 0, 6, -4, 9], [22, -4, 1, -3, -2, -1, 0, 0, 3, -5], [2, 4, 11, -2, 2, -3, -7, 2, 1, 6], [5, 2, -9, -6, -4, 1, 7, -9, 3, -8], [2, -7, 2, -2, -1, 4, -1, -3, -3, -8], [18, 0, -3, -4, -8, -2, -5, 4, 0, 2], [5, -3, 3, 4, -5, -10, -3, 7, 9, 8], [-17, 9, -9, -1, -4, 3, 3, -4, 4, -14], [6, 2, 4, 2, 16, 16, -6, -4, -2, -4], [11, 2, -3, -3, -9, -17, -5, 2, 15, 6], [14, 6, -6, -3, -5, -1, 2, -10, 4, -4], [0, -2, 0, 6, 4, 11, -1, 3, 1, -8], [-19, 10, 8, 7, -4, -12, 8, 14, 10, 5], [-6, 2, 0, 3, 9, 7, -2, -5, -5, -12], [-9, 5, 2, 1, 9, 7, -11, 6, -11, -1], [-8, 11, 4, 4, 6, 11, -9, 1, 0, 7], [4, 4, 3, 6, -1, -11, 5, 3, 8, 5], [4, 4, 5, 1, 13, 6, -8, 3, -14, 0], [26, 0, 3, -1, -4, -10, 2, -3, 10, 10], [-10, -13, -5, -9, -1, -6, 9, -5, -4, -11], [0, 2, -4, -2, -3, -13, 1, 2, 4, -1], [-3, 12, 2, 7, 4, 16, -3, -3, -2, -5], [-2, 6, 9, 3, 14, 7, -6, -2, -1, 13], [16, 5, -1, -7, -5, 0, -7, 1, -1, 3], [6, -4, 7, -4, -2, -13, 6, 3, 6, 9], [-24, -4, 1, 3, 2, -10, -2, 3, -5, 5], [4, 3, -7, 1, -12, -7, 8, 0, 5, -4], [5, -7, -6, 3, -5, -9, 10, 4, 2, -3], [-29, -1, -8, 0, -9, -14, 15, 0, 10, -13], [26, -1, 6, -3, -3, 1, 0, 2, 1, 3], [1, 0, 0, 2, 2, -14, 1, 1, 17, 6], [23, -1, -3, -8, 8, 4, 3, -8, 0, 3], [5, 1, -1, 10, 6, 13, 1, -1, -4, -11], [14, -3, -12, -7, -14, -8, 5, -7, 13, -7], [-3, 7, 5, 5, 3, 5, -5, -1, 2, 4], [-8, 1, -10, 0, -9, -7, 7, -8, 11, -8], [7, -9, -8, -5, -4, 1, -9, -3, 0, 2], [11, -5, 2, -1, -5, -9, -6, 5, 5, 5], [-15, -4, 6, 3, -2, -5, 5, 1, 11, 7], [0, 3, -2, -4, -2, 2, 0, 6, 9, 3], [-13, -13, -9, 5, 3, 5, 0, 1, -9, -2], [2, 1, 4, -2, 6, 2, -8, 9, -10, 1], [9, 10, 6, 6, -2, 8, -2, 1, 1, 6], [-25, -2, 2, -8, -4, -12, 3, 0, -3, -1], [-7, 1, 3, 5, 11, 0, 1, 7, -10, 5], [11, -2, -1, -1, 2, 1, -3, -8, 1, -5], [14, -11, 2, -4, 3, 6, -7, -8, -4, 0], [2, -13, 1, -4, -7, -10, 10, -1, 7, -4], [18, -3, -3, -6, -1, 8, -7, -5, -14, 4], [9, 7, -3, 3, 2, 13, 2, -2, -10, -10], [-9, -1, 2, 4, 0, -1, 6, 3, 4, -1], [24, 6, 12, 2, 2, -6, 2, 6, 8, 6], [-6, -2, 8, 0, 10, 10, -6, 4, -18, -4], [-3, 1, -13, -6, -10, 2, 1, -4, 10, -3], [-6, -8, 0, -1, 1, 9, 6, -7, -3, 4], [4, -4, -8, -1, 10, 11, 1, -3, -5, -6], [16, -9, 1, -1, -6, -14, -1, 0, 13, 3], [16, -3, -4, -8, -10, -16, 2, 0, 17, -7], [-16, -2, 4, 4, 0, 0, 1, -3, 1, -2], [21, 0, 4, 0, 6, -3, 2, 5, 5, 4], [2, 1, 4, 1, -1, 0, -2, 1, -5, 7], [2, -2, 8, 1, -3, -22, 7, 7, 11, 6], [-2, 0, 2, 3, -7, -8, 4, 2, 6, -8], [21, 2, 11, 10, 10, 16, -10, 4, -3, 8], [-14, 8, 0, 7, 11, 18, -15, 1, -13, 0], [-1, 7, -10, -6, -8, 3, -5, -6, 12, 5], [20, -4, 4, -5, -5, -11, 9, 2, 1, 4], [-2, -9, -8, 2, -6, -7, 5, 5, 8, 1], [1, 0, 8, -4, 10, 4, -10, -4, -6, 1], [-6, -3, -4, -2, -16, -2, -1, 3, -2, 2], [13, 7, 8, 7, 1, -3, -8, 6, 2, -3], [-18, -4, 8, 10, 12, 26, -14, 7, -16, 6], [-5, 3, -11, -6, -5, 5, -6, -7, -14, -8], [11, 7, 2, 0, 9, -1, -9, 0, -12, -4], [-10, 6, -6, -4, 0, -9, 15, 0, 2, -8], [-2, 19, -5, 4, 0, -1, -3, 0, 10, 5], [27, -2, -1, -7, 8, 14, -5, -13, -7, 1], [8, 6, 1, -1, 7, 6, 1, -2, -10, -4], [-3, 4, -5, -6, -8, 6, 0, -3, -3, 0], [9, -3, -5, -2, 8, 17, 1, -7, -8, -9], [25, 4, 2, -3, 6, 6, -7, -2, -3, 5], [19, -14, 7, -3, -4, -11, 7, 4, 4, 3], [-22, 4, -9, 2, -9, 4, 12, 3, 1, -6], [9, 13, 3, -4, 1, -8, 2, -3, -4, 4], [9, 7, 4, -6, 3, -9, 2, -3, 0, -6], [-14, 4, -4, -7, 0, -10, 5, -1, -1, 7], [21, -1, -6, 2, -4, 5, 0, -3, -9, -10], [22, -2, -1, -2, 9, 8, 0, -8, -8, 2], [-10, 3, -3, 7, -9, -2, -4, 8, 7, 5], [-10, 2, -10, -1, 6, 5, 3, 3, -6, -11], [12, -3, 4, -2, 6, 6, 5, -2, -12, 5], [11, -24, -1, -2, -3, -1, 1, -3, -8, -6], [0, 2, 1, 4, 5, 2, -1, -2, 4, 7], [11, -4, 1, 5, -3, -3, -7, -3, 12, 6], [10, -12, 7, 8, 14, 16, -6, 0, -14, -2], [16, -7, 7, 2, -3, -10, 2, -7, 13, 12], [10, 2, 3, -6, -6, -18, 8, -6, 13, 0], [-22, -4, -1, 6, -13, -16, 1, 9, -1, -9], [-2, -14, -15, -9, -15, -3, 10, -6, 3, -8], [17, 6, 6, -5, -4, -9, 0, 4, 10, 7], [0, 9, -2, 3, 1, -16, 17, -1, 12, 1], [-19, -3, 5, 6, -3, -8, 6, 16, -4, 4], [-6, -7, -15, -3, -14, -6, 7, -3, -4, -11], [0, -5, -1, 0, 3, 8, 3, -11, -6, -16], [13, 4, -3, -2, 4, 7, -2, -5, -4, -4], [-6, -16, -19, -3, -7, 4, 8, -8, -2, -12], [1, -9, -5, 4, -9, -7, 16, -2, 16, -8], [18, 2, 3, -7, 8, 6, -2, -9, -6, 3], [-4, 4, -17, -3, 3, 20, 3, -7, -12, -16], [20, 4, 7, 0, 12, 9, -5, -3, 0, 0], [7, -14, -8, -6, 2, 4, -8, -2, -10, -5], [2, -10, -10, -5, -8, -1, 12, -12, 5, -3], [-9, -2, -9, 0, -6, 4, 8, 0, 2, -14], [9, -4, -4, 0, 2, -4, -6, -1, 6, 10], [-12, -4, -1, 1, 6, -2, -2, 6, -9, -9], [-4, 6, -3, -10, -6, 0, -2, -8, 5, 2], [15, -7, -5, -3, -5, -15, 8, 3, 2, 3], [25, 0, -5, -6, -11, -18, 2, -1, 17, 9], [4, -14, 3, 2, 10, 5, -1, -1, -16, -2], [-22, -5, 2, 6, 2, 4, -1, -4, -18, -7], [19, -2, 1, 16, 8, 21, -4, -1, -6, -12], [-3, -8, 10, 6, 19, 12, -4, -5, -7, 11], [-22, -4, 4, 11, -9, -3, 0, 13, 6, 6], [15, -15, 8, 1, 2, -4, 2, -2, -4, -4], [-19, 5, -10, -10, -14, -15, 11, -1, -1, -5], [-5, -6, 0, 2, 7, 5, 4, 3, -13, 1], [10, 4, 18, 7, 3, -9, -10, -2, 1, 7], [12, 7, -5, -2, -17, -12, -1, 8, 6, 2], [26, 1, -5, -4, 9, 9, -4, 2, -12, -1], [2, -7, -2, -1, -4, 4, -8, 5, -5, -6], [4, -12, -5, 1, 7, 11, -4, 4, -15, -2], [-10, -13, 2, 3, 2, 7, -4, -1, -14, -10], [13, -2, 4, 1, 4, 13, -19, 3, -5, 15], [-18, 2, 12, 2, -6, -14, 9, 7, 5, -1], [10, -12, 6, -12, -2, -8, 10, -2, 4, 8], [1, -13, 9, 5, -8, -9, 11, 6, 14, -4], [10, 9, 15, 1, 14, 7, -3, -1, -8, 12], [-11, -10, 1, 11, 15, 15, -11, -4, -13, 4], [-8, -11, -9, -2, -16, -6, 13, -7, 18, -7], [-5, 13, -6, -5, -12, -11, 7, 3, 22, 2], [17, 5, 5, -11, 7, 16, -11, -12, -3, 5], [-9, 7, 12, 12, -3, -12, -1, 16, 7, 14], [-4, -2, -9, -9, -7, -16, 15, -5, 6, -6], [15, 6, 7, -1, -5, -6, 2, 5, 4, 18], [-16, 13, -1, -10, -8, -4, -1, 1, 8, -1], [5, -5, -15, -6, -17, 4, 6, -6, -1, -12], [-7, -5, 2, 4, -7, 5, -6, 1, 3, 10], [-2, 2, 13, 1, -9, -7, -8, 3, 4, 6], [5, 4, 0, -7, -9, -7, -3, -6, 18, 10], [20, -9, 0, 1, 4, -7, 7, 1, 21, -4], [2, -15, -2, -3, -7, -5, 8, 2, 9, -9], [2, 7, -2, -12, 5, -14, 17, -4, 6, 0], [-27, -4, 12, -2, 2, -11, 3, 2, 3, 4], [24, -11, -6, -6, 6, 2, -15, -7, 1, 5], [4, 4, 17, 6, 14, -6, -8, 2, -8, 17], [-19, -8, -6, 1, 3, 8, 3, -5, -5, -4], [-2, 2, 7, -2, -2, -4, 2, -5, -6, -10], [3, 2, -6, 4, 7, 14, -12, -5, -1, 4], [-12, 19, -9, -11, -9, -8, 3, -4, 8, -7], [-4, -3, -15, -10, -6, 11, -7, -6, -14, -1], [1, -6, -6, 2, -5, -15, 3, 1, 22, -1], [15, -5, -8, -3, 15, 25, 1, -6, -13, -11], [-25, 14, 1, 4, 5, 17, -2, -4, -12, -6], [-1, -18, -5, -3, -8, -5, -4, -1, -1, -5], [5, 9, 8, 3, 10, -2, -4, 0, 3, 14], [-8, 0, 4, -4, -3, -3, 4, 6, 2, 0], [-6, 17, 11, 10, 9, 10, -4, 4, 3, -2], [13, 0, 5, 6, 0, -12, 7, -1, 6, 2], [4, -10, -3, -8, -5, -10, 6, 3, 4, -11], [-12, -9, 2, 6, 6, -1, 4, -3, 5, 5], [-5, 8, -10, -8, -6, 15, -5, -9, -17, -10], [-19, 5, -9, 5, -11, -20, 8, 9, 9, -5], [7, 8, 3, -9, -7, -3, 3, -11, 3, 0], [-11, -2, -5, -2, 10, -5, 9, -4, 0, -4], [2, -4, 13, 4, 11, 1, -11, 6, 1, 19], [0, -7, 0, 2, 1, -2, -11, 3, 14, 12], [-2, -1, -17, -5, -19, 2, 4, -3, 10, 1], [3, -4, -1, -8, 2, 9, -5, -18, -2, -6], [13, -20, -12, -4, 2, 9, 4, -3, -13, -16], [16, -8, 3, -12, -4, -3, -5, -12, 6, 13], [8, 10, -16, -8, -7, 9, -6, -15, -8, -15], [-4, -9, -8, 5, 3, 15, -1, -4, -14, -3], [-16, 1, -6, -3, 1, 3, -7, -1, 3, 13], [-6, 0, 14, 8, 0, -13, 7, 12, 1, 15], [-20, 4, -11, -2, -7, -4, 2, 7, -10, -9], [-25, 1, 7, 3, 17, 16, -11, 11, -12, 3], [22, -4, -1, -5, 12, 10, -10, -3, -17, -5], [8, -7, 2, 4, -7, 2, 9, 8, 6, 4], [-4, -5, -5, -3, -18, -2, 6, -6, 0, 0], [-7, 21, 10, -2, 0, 9, -6, -7, -2, -1], [20, -5, 14, -2, 4, -6, 2, 9, 11, 11], [21, -10, 7, 1, 4, -9, -1, -2, 3, 15], [-23, -3, 8, 14, 6, 2, -2, 5, 0, -1], [2, 5, -5, -6, -19, -28, 3, 7, 14, 10], [16, 4, 2, 3, -3, -6, 9, -3, 3, -26], [18, -5, -3, 0, 3, 3, 2, -2, -1, -6], [2, 9, 19, 6, 11, 0, -2, 8, 14, 10], [-18, 0, -13, -4, -19, -5, 4, 6, -5, -2], [1, -1, 0, 1, -6, 0, 5, -3, -5, 8], [16, 11, -12, 3, 13, 32, -16, -12, -18, 1], [-2, 9, -6, 5, 5, 15, -11, 0, -8, -5], [-8, -12, -9, -4, -15, 4, 3, -6, 6, -5], [9, 0, -21, -9, 0, 10, 4, -7, -8, -11], [7, 5, -7, -11, -1, 2, 3, -3, 9, 0], [-9, -14, -1, 9, 1, 24, -9, -7, -7, 0], [-8, -11, 14, 10, -6, 0, -2, -6, 7, 7], [2, -10, 2, 2, 12, 2, -2, 0, -17, 2], [4, -6, -7, -7, -6, -11, 18, -6, 5, -13], [23, 8, 17, 14, 7, 11, -11, 3, 4, 10], [-8, 6, 3, -3, 6, -11, -3, 7, 3, -3], [5, -13, 8, 17, 13, 18, -6, -1, -4, -1], [-24, -1, 6, 6, 6, 14, -9, 7, -18, -9], [-7, -2, 10, 0, 1, -9, 10, 1, 7, -5], [-12, 6, 3, 12, 5, -3, -2, 20, 0, -7], [30, 6, 5, 10, 13, 19, -2, 5, -13, -7], [25, -5, 9, -1, 11, 8, -2, 9, -7, 13], [23, 2, 0, -1, -9, -21, 2, -1, 12, 10], [27, -1, -1, 6, 12, 9, -4, -6, -4, -3], [-4, -3, 4, -6, -13, -19, -7, 4, 21, 16], [-3, -9, 3, 5, -18, -11, -3, 1, 9, 6], [-22, 10, -10, 6, -8, 1, 7, 0, 7, -6], [11, 4, -16, -11, -8, 4, -2, -11, 9, 3], [3, 12, -1, -6, 3, 4, -13, -5, 5, 12], [10, -16, 0, 12, -8, -14, -2, 2, 8, 0], [-12, -5, -9, 2, 6, 20, 6, -1, -14, -9], [-25, 9, 1, 2, -2, 9, -8, 5, -8, 5], [17, -1, -2, -6, 6, 2, -3, -9, 9, 0], [-2, 2, -13, -9, -5, -10, 11, -7, -8, -18], [-28, -3, -4, 10, 7, 10, -3, 6, -14, -16], [11, -20, -6, -4, 2, 7, 10, -10, -9, -12], [-3, -2, -9, -1, 14, 2, -1, 9, -2, -4], [13, 9, 5, 1, 15, 18, -2, -3, -20, 0], [-11, 11, -5, 6, 1, 10, -9, -1, 3, 6], [2, 10, 3, 2, -7, -18, 6, -5, 29, 18], [-19, 2, 6, 7, 1, -14, 4, 11, 3, 2], [-22, 5, 9, 11, 12, -8, -2, 15, 0, 10], [4, -11, -8, 7, -8, -12, 2, 12, 4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_8034_2_a_s(:prec:=10)
    chi := MakeCharacter_8034_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8034_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_8034_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![256, 112, -2392, 1671, 1013, -945, -49, 151, -17, -6, 1]>,<7,R![-3088, 5417, 628, -4323, 540, 1207, -123, -162, 0, 9, 1]>],Snew);
    return Vf;
end function;